Alternating Minimization Multigrid Algorithms for Transmission Tomography

Transcription

1 Januar SPI Coputatonal Iagng II 1 Alternatng Mnzaton Multgrd Algorths for Transsson Toograph Joseph A. O Sullvan and Jasenka Benac lectronc Sstes and Sgnals Research Laborator Departent of lectrcal and Sstes ngneerng Washngton Unverst jao@wustl.edu Ths work was supported n part b NIH grant R01CA75371 fro the Natonal Cancer Insttute J. F. Wllason VCU PI.

2 Januar SPI Coputatonal Iagng II 2 Outlne Motvaton Transsson toograph n presence of hgh denst aterals Alternatng Mnzaton algorths Multgrd AM Prelnar Results Conclusons

3 Januar SPI Coputatonal Iagng II 3 CT Iagng n Presence of Hgh Denst Attenuators Brachtherap applcators: after-loadng colpostats for radaton oncolog. Dose predcton portant Truth FBP Fltered backprojecton: nverse Radon transfor. ffects of nose scatter bea hardenng pronent due to hgh attenuators.

4 Januar SPI Coputatonal Iagng II 4 Transsson Toograph Source-detector pars ndeed b pels ndeed b Data d Posson eans g:µ log lkelhood functon Mean unattenuated counts I 0 ean background β Attenuaton functon µ energes Maze over µ or c equvalentl nze I-dvergence ep : : : ln : 0 h I g g g d g d l β µ µ µ µ µ + X Y N c 1 µ µ

5 Januar SPI Coputatonal Iagng II 5 + N p q d p p c h I q q p q p p q p I : ep ln n n 1 0 L L X Y µ Lnear fal deterned b the easured data. ponental fal paraeterzed b ages. Alternatng Mnzaton Algorth Iage foraton stateent

6 Januar SPI Coputatonal Iagng II 6 Alternatng Mnzaton Algorth Iage update ˆ ~ ln 1 ˆ ˆ 1 b b Z c c l l l l + Interpretaton: copare predcted data to easured data va rato of backprojectons update estate usng a noralzaton constant l l l l q h b p h b ˆ ˆ ˆ ~ µ µ

7 Januar SPI Coputatonal Iagng II 7 Alternatng Mnzaton Algorth cˆ l+ 1 Iage update ~ l 1 b cˆ ln Z bˆ Interpretaton: copare predcted data to easured data va rato of backprojectons update estate usng a noralzaton constant Coents: choce for constants onotonc convergence constrants easl ncorporated coputatonall epensve: N forward 2 N backward projectons per teraton l l

8 Januar SPI Coputatonal Iagng II 8 Increased Convergence Rates Ordered subsets Hudson and Larkn; Kaphus and Beekan; etc. Separable Parabalodal Surrogate functons Fessler et al. Multgrd Methods Bouan et al Fast forward/backward projectons Bresler et al. Other see Sotthvrat Ahn Fessler

9 Januar SPI Coputatonal Iagng II 9 Multgrd Approach Oh Mlsten Bouan Webb et al. Objectve functons at ultple grds Surrogate functon vew: atch value and gradent at current estate cost + 1 cost J J + 1 Our pleentatons: lttle speedup; not atched to AM approach not onotonc; negatve values n estates nsde of logarths + 1 cost cost J

10 Januar SPI Coputatonal Iagng II 10 Multgrd AM Algorth Approach: copute age correcton c on coarse grd usng AM c c c [0] J + c 0 0 c [ ] 0 J Fne grd Interpolatng operator 0 1 2

11 Januar SPI Coputatonal Iagng II 11 Multgrd AM Algorth [ ] + + X Y N p q c J c h I q q p q p p q p I 1 ] [ 0 [0] 0 ep ln n n µ L Coents: onotonct guaranteed faster coputatons on coarser grds fleble grd sequence

12 Januar SPI Coputatonal Iagng II 12 Multgrd AM Algorth Iteraton overvew + k k k k k k k k c h I q J h c J h q b J h d b K k b b c c ˆ ep ˆ ˆ ep ˆ ˆ 1 0 ˆ ln Z 1 ˆ ˆ 0 0 ] [ 0 ] [ 0 ] [ ] [ ] [ 1 ] [ Κ On each grd run K teratons: Fne grd age update: ˆ ˆ ˆ ] [ 0 1 c J c c K k k + +

13 Januar SPI Coputatonal Iagng II 13 Prelnar Results Sulated data: Lucte core n water bath wth four etallc nserts 4 grd levels Interpolaton average of four neghbors Coent: potental satch n projecton operators

14 Januar SPI Coputatonal Iagng II 14 Conclusons and Future Work Multgrd Alternatng Mnzaton Algorth potental for ncreasng convergence rate of sngle grd AM guaranteed onotonc convergence propertes Future analss Decaton operators n age space Incorporate decaton n easureent space Ordered subsets and ultgrd cobned

15 Januar SPI Coputatonal Iagng II 15 References J. A. O Sullvan and J. Benac Alternatng nzaton algorths for transsson toograph subtted to I Trans. Med. Ig. S. Oh B. Mlsten C. A. Bouan K. J. Webb Adaptve nonlnear ultgrd nverson wth applcatons to Baesan optcal dffuson toograph n Proc. I Stat. Sgnal Processng Workshop 2003 T.-S. Pan A.. Yagle Nuercal stud of ultgrd pleentatons of soe teratve age reconstructon algorths n Proc. I Nuclear Sc. Sposu and Med. Ig. Conference Z. Wu A. H. Tewfk Multgrd algorth for age reconstructon fro Fourer odulus n Proc. ICASSP 1991.

16 Januar SPI Coputatonal Iagng II 16 Mn CT AM Iteraton Davd G. Poltte October

17 Januar SPI Coputatonal Iagng II 17 Mn CT AM Iteraton Davd G. Poltte October

18 Januar SPI Coputatonal Iagng II 18 Mn CT AM Iteraton Davd G. Poltte October

19 Januar SPI Coputatonal Iagng II 19 Mn CT AM Iteraton Davd G. Poltte October

20 Januar SPI Coputatonal Iagng II 20 Mn CT AM Iteraton Davd G. Poltte October

21 Januar SPI Coputatonal Iagng II 21 Mn CT AM Iteraton Davd G. Poltte October

22 Januar SPI Coputatonal Iagng II 22 Mn CT AM Iteraton Davd G. Poltte October

23 Januar SPI Coputatonal Iagng II 23 Mn CT AM Iteraton Davd G. Poltte October

24 Januar SPI Coputatonal Iagng II 24 Mn CT AM Iteraton Davd G. Poltte October

25 Januar SPI Coputatonal Iagng II 25 Mn CT AM Iteraton Davd G. Poltte October

26 Januar SPI Coputatonal Iagng II 26 Mn CT AM Iteraton Davd G. Poltte October

27 Januar SPI Coputatonal Iagng II 27 Mn CT AM Iteraton Davd G. Poltte October

28 Januar SPI Coputatonal Iagng II 28 Mn CT AM Iteraton Davd G. Poltte October

29 Januar SPI Coputatonal Iagng II 29 Mn CT AM Iteraton Davd G. Poltte October

30 Januar SPI Coputatonal Iagng II 30 Mn CT AM Iteraton Davd G. Poltte October

31 Januar SPI Coputatonal Iagng II 31 Mn CT AM Iteraton Davd G. Poltte October

32 Januar SPI Coputatonal Iagng II 32 Mn CT AM Iteraton Davd G. Poltte October

33 Januar SPI Coputatonal Iagng II 33 Mn CT AM Iteraton Davd G. Poltte October

34 Januar SPI Coputatonal Iagng II 34 Mn CT AM Iteraton Davd G. Poltte October